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The Banach–Tarski paradox was a proof that you can break a sphere into a finite number of parts (as few as 5 parts), rearrange the pieces by only moving them around (not changing their size at all), and getting two whole spheres out of the process. It doesn't apply in the real world because it would require infinitely small and many cuts to make the parts, and the parts have a property called "nonmeasurable" meaning they can't have a defined value for their volume. Measure theory alone can give headaches to math undergrads, which applies to things that are measurable, so the math behind the paradox can be quite esoteric. But part of the point was to demonstrate that geometric intuition doesn't apply if you can make infinitely complicated cuts and divisions to things.